Construction of Lyapunov Certificates for Oscillatory Systems

We develop a method for constructing Lyapunov functions via Semidefinite Programming (SDP) that certifies the stability of oscillatory systems with both Cartesian and angular variables. We utilize the theory of hybrid polynomials (also called mixed trigonometric-polynomials) introduced by Dumitrescu. We use this theory to convert Lyapunov and dual Lyapunov stability conditions for oscillatory systems into SDP problems. Solving these problems using standard convex programming solvers leads to expressions of Lyapunov densities and local Lyapunov functions for these systems, even without apriori knowing the invariant attracting set. To illustrate the applicability of our method, we consider the analysis of Kuramoto models and the state feedback design problem for an inverted pendulum on a cart. Specifically, we establish certificates of almost global synchronization (phase locking) for second-order Kuramoto models. The paper concludes by developing an SDP certificate that enables the design of a swing-up control for an inverted pendulum on a cart. For the analysis, we use our program vSOS-hybrid, based on CVX in MATLAB, openly available on GitHub.